# Subrings and multiplicative identities

If a ring has a 1 (i.e. a ring has a multiplicative inverse), does this imply that the subring has a 1 as well?

Thank you

• multiplicative identity? Look at $\Bbb Z$ and $2 \Bbb Z$ – Chinnapparaj R Apr 1 at 15:15
• Think about the zero divisors. – Μάρκος Καραμέρης Apr 1 at 15:15
• Yes. Subrings of rings with identity contain the multiplicative identity. There are authors that say otherwise so, ultimately, you have to consult your textbook or your teacher. I disagree with those authors because a sub$thing$ must preserve the structure of the $thing$. Since $1$ is part of the structure of a ring with identity it should be preserved by subrings. – John Douma Apr 1 at 15:21

If your definition of "subring $$S$$ of $$R$$" includes verbiage saying "and it has to have the same identity as $$R$$" then trivially yes.
An abelian subgroup that is closed under multiplication could have no identity element, or it could have an identity distinct from that of $$R$$, or it could have the same identity as $$R$$. Which one of these you call "a subring" is up to the definitions you've settled on.